Automatic differentiability and characterization of cocycles of holomorphic flows

Jafari, Farhad; Tonev, Thomas; Toneva, Elena

doi:10.1090/s0002-9939-05-07904-9

Cited by 4 publications

(3 citation statements)

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“…For A commutative, we show that the answer is positive, and an explicit expression for the mapping M is constructed in terms of the 'generator' of the semicocycle. The proof of this result, which slightly generalizes the results of [8] for the case A = C, relies very strongly on commutativity. In fact, as soon as we move to the non-commutative case, a simple example with A = M 2 (C) shows that there are semicocycles which are not linearizable (see Example 3.2).…”

Section: Introductionsupporting

confidence: 58%

“…Several works have studies holomorphic semicocycles over semigroups of holomorphic self-mappings of the open unit disk D of the complex plane C, with values in C, which are commutative semicocycles. In particular, it was shown in [8] (see also [11]) that each C-valued semicocycle is smooth; the question whether a semicocycle is a coboundary was considered in [11,9].…”

Section: Introductionmentioning

confidence: 99%

“…For question (Q1), we will give a positive answer in Section 2, proving that every semicocycle is generated by an appropriate holomorphic mapping B (see Theorem 2.2 below). The main step here is to prove that a semicocycle is automatically differentiable with respect to its parameter t. For the case A = C, this was proved in [8], but the proof given there does not seem easy to generalize to the non-commutative case. We therefore develop an alternative proof, using an approach which appears to us simpler, even in the case A = C. In this section, we also apply Theorem 2.2 to derive growth estimates for semicocycles.…”

Section: Introductionmentioning

confidence: 99%

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Noncommutative Holomorphic Semicocycles

Elin¹,

Jacobzon²,

Katriel³

2019

Michigan Math. J.

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This paper studies holomorphic semicocycles over semigroups in the unit disk, which take values in an arbitrary unital Banach algebra. We prove that every such semicocycle is a solution to a corresponding evolution problem. We then investigate the linearization problem: which semicocycles are cohomologous to constant semicocycles? In contrast with the case of commutative semicocycles, in the non-commutative case non-linearizable semicocycles are shown to exist. Simple conditions for linearizability are derived and are shown to be sharp.

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Section: Introductionsupporting

confidence: 58%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Noncommutative Holomorphic Semicocycles

Elin¹,

Jacobzon²,

Katriel³

2019

Michigan Math. J.

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Semigroups of Operators on Hardy Spaces and Cocycles of Holomorphic Flows

Jafari

Słodkowski

Tonev

2010

Complex Anal. Oper. Theory

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Weighted Composition Semigroups on Spaces of Continuous Functions and Their Subspaces

Kruse

2024

Complex Anal. Oper. Theory

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This paper is dedicated to weighted composition semigroups on spaces of continuous functions and their subspaces. We consider semigroups induced by semiflows and semicocycles on Banach spaces $$\mathcal {F}(\Omega )$$ F ( Ω ) of continuous functions on a Hausdorff space $$\Omega $$ Ω such that the norm-topology is stronger than the compact-open topology like the Hardy spaces, the weighted Bergman spaces, the Dirichlet space, the Bloch type spaces, the space of bounded Dirichlet series and weighted spaces of continuous or holomorphic functions. It was shown by Gallardo-Gutiérrez, Siskakis and Yakubovich that there are no non-trivial norm-strongly continuous weighted composition semigroups on Banach spaces $$\mathcal {F}(\mathbb {D})$$ F ( D ) of holomorphic functions on the open unit disc $$\mathbb {D}$$ D such that $$H^{\infty }\subset \mathcal {F}(\mathbb {D})\subset \mathcal {B}_{1}$$ H ∞ ⊂ F ( D ) ⊂ B 1 where $$H^{\infty }$$ H ∞ is the Hardy space of bounded holomorphic functions on $$\mathbb {D}$$ D and $$\mathcal {B}_{1}$$ B 1 the Bloch space. However, we show that there are non-trivial weighted composition semigroups on such spaces which are strongly continuous w.r.t. the mixed topology between the norm-topology and the compact-open topology. We study such weighted composition semigroups in the general setting of Banach spaces of continuous functions and derive necessary and sufficient conditions on the spaces involved, the semiflows and semicocycles for strong continuity w.r.t. the mixed topology and as a byproduct for norm-strong continuity as well. Moreover, we give several characterisations of their generator and their space of norm-strong continuity.

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Automatic differentiability and characterization of cocycles of holomorphic flows

Cited by 4 publications

References 5 publications

Noncommutative Holomorphic Semicocycles

Noncommutative Holomorphic Semicocycles

Semigroups of Operators on Hardy Spaces and Cocycles of Holomorphic Flows

Weighted Composition Semigroups on Spaces of Continuous Functions and Their Subspaces

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